Mathematics for the interested outsider

Measures

From this point in, I will define a “set function” as a function whose domain is some collection of subsets . It’s important to note here that is not defined on points of the set , but on subsets of . For some reason, a lot of people find that confusing at first.

We’re primarily concerned with set functions which take their values in the “extended real numbers” . That is, the value of is either a real number, or , or , with the latter two being greater than all real numbers and less than all real numbers, respectively.

We say that such a set function is “additive” if whenever we have disjoint sets and in with disjoint union also in , then we have

Similarly, we say that is finitely additive if for every finite, pairwise disjoint collection whose union is also in we have

And we say that is countably additive of for every pairwise-disjoint sequence of sets in whose union is also in , we have

Now we can define a “measure” as an extended real-valued, non-negative, countably additive set function defined on an algebra, and satisfying . With this last assumption, we can show that a measure is also finitely additive. Indeed, given a collection , just define for to get a sequence. Then we find

If is a measure on , we say a set has finite measure if . We say that has “-finite” measure if there is a sequence of sets of finite measure () so that . If every set in has finite (or -finite) measure, we say that is finite (or -finite) on .

Finally, we say that a measure is “complete” if for every set of measure zero, also contains all subsets of . That is, if , , and , then . At first, this might seem to be more a condition on the algebra than on the measure , but it really isn’t. It says that to be complete, a measure can only assign to a set if all of its subsets are also in .

[…] we don’t particularly care if the set where is false is itself measurable, although if is complete then all -negligible sets will be measurable. This sort of language is so common in measure theory […]

[…] real-valued functions on them. Given a function on a Boolean ring , we say that is additive, or a measure, -finite (on -rings), and so on analogously to the same concepts for set functions. We also say […]

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.